Passaggia
Pierre-Yves
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Optimal transient growth in thin-interface internal solitary waves
Passaggia
Pierre-Yves
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Preprint
The dynamics of perturbations to large-amplitude Internal Solitary Waves (ISW) in
two-layered
ows with thin interfaces is analyzed by means of linear optimal transient
growth methods. Optimal perturbations are computed through direct-adjoint iterations
of the Navier-Stokes equations linearized around inviscid, steady ISWs obtained from
the Dubreil-Jacotin-Long (DJL) equation. Optimal perturbations are found as a function
of the ISW phase velocity c (alternatively amplitude) for one representative stratification. These disturbances are found to be localized wave-like packets that originate
just upstream of the ISW self-induced zone (for large enough c) of potentially unstable
Richardson number, Ri < 0:25. They propagate through the base wave as coherent
packets whose total energy gain increases rapidly with c. The optimal disturbances are
also shown to be relevant to DJL solitary waves that have been modi ed by viscosity
representative of laboratory experiments. The optimal disturbances are compared to the
local WKB approximation for spatially growing Kelvin-Helmholtz (K-H) waves through
the Ri < 0:25 zone. The WKB approach is able to capture properties (e.g., carrier frequency,
wavenumber and energy gain) of the optimal disturbances except for an initial
phase of non-normal growth due to the Orr mechanism. The non-normal growth can
be a substantial portion of the total gain, especially for ISWs that are weakly unstable
to K-H waves. The linear evolution of Gaussian packets of linear free waves with the
same carrier frequency as the optimal disturbances is shown to result in less energy gain
than found for either the optimal perturbations or the WKB approximation due to nonnormal
effects that cause absorption of disturbance energy into the leading face of the
wave. Two-dimensional numerical calculations of the nonlinear evolution of optimal disturbance
packets leads to the generation of large-amplitude K-H billows that can emerge
on the leading face of the wave and that break down into turbulence in the lee of the wave.
The nonlinear calculations are used to derive a slowly varying model of ISW decay due
to repeated encounters with optimal or free wave packets. Field observations of unstable
ISW by Moum et al. (2003) are consistent with excitation by optimal disturbances.